(Ultra)differentiable functional calculus and current extension of the resolvent mapping

Andersson, Mats

Annales de l'Institut Fourier, Tome 53 (2003) no. 3, pp. 903-926.

Soit $a=(a_1,...,a_n)$ un $n$-uplet d’opérateurs commutant entre eux sur un espace de Banach $X$. Nous discutons diverses conditions équivalentes pour que le calcul fonctionnel holomorphe (de Taylor) s’étende aux fonctions analytiques réelles ou à diverses classes ultra-différentiables. En particulier, nous abordons la possibilité d’un calcul fonctionnel pour les fonctions lisses, qui est liée à l’existence d’un prolongement de l’application résolvante en tant que (ultra-)courant sur le spectre (de Taylor) de $a$. Si $a$ est un $n$-uplet admettant un calcul fonctionnel lisse on peut définir une opération de translation par $a$ sur les fonctions lisses (et les formes) à valeurs dans $X$. Nous obtenons comme application une nouvelle démonstration simple de la propriété ${\left(\beta \right)}_{ℰ}$.L’outil principal que nous introduisons dans cet article, et dont nous pensons qu’il a un interêt propre, est la transformée de Fourier des formes et des courants. Nous en démontrons quelques propriétés fondamentales telles que la formule d’inversion et calculons la transformée de Fourier de certains courants particuliers.

Let $a=(a_1,...,a_n)$ be a tuple of commuting operators on a Banach space $X$. We discuss various conditions equivalent to that the holomorphic (Taylor) functional calculus has an extension to the real-analytic functions or various ultradifferentiable classes. In particular, we discuss the possible existence of a functional calculus for smooth functions. We relate the existence of a possible extension to existence of a certain (ultra)current extension of the resolvent mapping over the (Taylor) spectrum of $a$. If $a$ is a tuple that admits a smooth functional calculus we can define an operation translation by $a$ on $X$-valued smooth functions (and forms). As an application we get a new simple proof of the so-called ${\left(\beta \right)}_{ℰ}$ property.The main tool that we introduce in this paper, and which we think has an independent interest, is Fourier transforms of forms and currents. We prove some basic properties including the inversion formula and compute the Fourier transforms of some special currents.

Classification : 47A60, 47A13, 32A25
Keywords: commuting operators, generalized scalar operator, functional calculus, Bishop’s property $(\beta )$, Taylor spectrum, ultradifferentiable function, resolvent mapping, current
Mot clés : opérateurs commutants, opérateur scalaire généralisé, calcul fonctionnel, propriété $(\beta )$ de Bishop, spectre de Taylor, fonction ultradifférentiable, application résolvante, courant

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